A reference is Rudin, Principles of mathematical analysis, Example 9.43. Here the integral
$$
\int_{-\infty}^{\infty}e^{-x^2}\cos(xt)\,dx
$$
is calculated using the theory of ordinary differential equations.
The integral is
$$
\sqrt{\pi}\exp\left(-\frac{t^2}{4} \right).
$$
(Hint) In your integral after introducing new variable you should calculate
$$
\int_{-\infty}^{\infty}e^{-z^2}\cos(cz)\,dz
$$
and
$$
\int_{-\infty}^{\infty}e^{-z^2}\sin(cz)\,dz.
$$